In one of our Brain Potions from last year (found here), we explored a betting strategy to supposedly win "Infinite Money" playing a version of Roulette.

In this week's Brain Potion we present a simple card game and explore how much money we can win!

Start with a deck of $2N$ cards, with $N$ black cards and $N$ red cards and start with $\$1.00$. The dealer of the game flips the cards over, one by one, until all $2N$ cards are dealt. Before he flips over each card, you have a chance to bet/predict the color of the next card. You can choose to risk some/all/none of your money, and if you are correct you win double what you bet.

Example: if you have $\$\dfrac{2}{3}$ and choose to bet $\$\dfrac{1}{6}$, then if you are wrong you end up with $\dfrac{2}{3} - \dfrac{1}{6} = \$\dfrac{1}{2}$ while if you are correct you end up with $\dfrac{2}{3} - \dfrac{1}{6} + \dfrac{2}{6} = \$\dfrac{5}{6}$.

Your goal is to come up with a strategy to guarantee you win as much money as possible, leaving nothing to chance.

Example: if $N=1$ we have one red $R$ and one black $B$ card. We can guarantee we finish with $\$2$ using the following strategy. Bet nothing before the first card is flipped. In this way you learn the result of the first card and can bet your full $\$1$ on the second card (and are guaranteed to be correct!). This gives you $1 + 1 = \$2$ at the end.

Try to come up with a strategy for $N = 2$. That means you have to have a strategy that deals will all possible ways the cards could be dealt:
$$BBRR, BRBR, BRRB, RBBR, RBRB, RRBB$$
so that you always end up with the same amount of money. Hint: You can guarantee you win more than $\$2$!

As a challenge, once you've solved the problem for $N=2$, try $N=3$, etc. There is a nice pattern for how much money you can win, depending on $N$. Try to find (and prove) it!